Polynomials: LMVT always applies (continuous and differentiable everywhere). For degree n polynomial, f'(c) = [f(b)-f(a)]/(b-a) gives a degree n-1 equation in c, which always has a solution in (a,b).

Exponentials: f(x) = $e^x$ on [0,1]. f'(c) = $e^c$ = e-1. c = ln(e-1) ≈ 0.541.

Trigonometric: f(x) = sin x on [0, pi]. f'(c) = cos c = $\frac{0-0}{pi}$ = 0. c = pi/2. The tangent at pi/2 is horizontal, parallel to the secant from (0,0) to (pi,0).

Logarithmic: f(x) = ln x on [1, e]. f'(c) = 1/c = $\frac{1-0}{(e-1)}$. c = e-1 ≈ 1.718.